We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $\Omega(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(\Omega(\sqrt{\log |V(G_n)|}))$ expand losslessly.

翻译：我们研究构造具有强顶点扩展性质的显式稀疏图问题。主要结果是首个双侧不平衡单邻扩展图构造，即二分图中左右两部分的小集合均呈现单邻扩展性，同时具备与量子码构造相关的代数性质。我们的构造通过将大规模三部图谱扩展图与足够好的常量尺寸单邻扩展图进行三部线积实现——这是我们定义的新图积，它推广了Alon与Capalbo工作中的线积以及Asherov与Dinur工作中的路由积。为分析三部线积所得图的顶点扩展性质，我们建立了一个关于二分图谱扩展图中可能出现的子图的精确刻画，该结果推广了Kahale的相关结论，可能具有独立研究意义。通过选取适当的图应用该图积，我们给出了无限族$(d_1,d_2)$-双正则图$(G_n)_{n\ge 1}$（对足够大的$d_1$和$d_2$）的强显式构造，其中所有顶点数少于小常数比例的子集$S$均具有$\Omega(d_1\cdot |S|)$个单邻（假设$d_1 \leq d_2$）。此外，我们还可保证规模不超过$\exp(\Omega(\sqrt{\log |V(G_n)|}))$的顶点子集实现无损扩展。